Quantitative recurrence for $\mathbb {Z}$-extension of three-dimensional Axiom A flows

Abstract In this paper, we study the quantitative recurrence properties in the case of $\mathbb {Z}$ -extension of Axiom A flows on a Riemannian manifold. We study the asymptotic behavior of the first return time to a small neighborhood of the starting point. We establish results of almost everywhere convergence, and of convergence in distribution with respect to any probability measure absolutely continuous with respect to the infinite invariant measure. In particular, our results apply to geodesic flows on the $\mathbb {Z}$ -cover of compact smooth surfaces of negative curvature.